Seismic design and retrofit of frame structures with hysteretic dampers: a simplified displacement-based procedure

Abstract

This paper describes an effective and easy to use displacement-based procedure for seismic design or retrofit of frame structures equipped with hysteretic dampers, taking into account the flexibility of the supporting brace that is usually provided to connect the device to the external frame. The proposed framework leads the designer to the definition of a complete set of dissipative braces mechanical properties able to provide a desired performance level. Some initial assumptions related to the equivalent damped brace system have to be set and checked throughout the procedure. The method is widely explained step by step, differentiating the case of linear elastic and nonlinear behavior of the bare frame. The capacity curve of the braced frame is built by means of simple analytical relations and approximated by a bilinear or trilinear curve depending on the bare frame behavior. Two case studies are provided to demonstrate the effectiveness of the suggested procedure for both cases of new construction and existing building, obtaining a satisfactory matching between analytical target and numerical capacity curves. The reliability of the design framework is finally assessed by means of static and dynamic nonlinear analyses.

Appendices

Appendix A: Proof of Eq. (8)

Making reference to Fig. 24, the following equalities apply:

$${\text{V}}_{\text{PP,BF}}^{ *} = {\text{K}}_{\text{BF}}^{ *} \cdot {\text{d}}_{\text{y}}^{ *} + {\text{K}}_{\text{BF,py}}^{ *} \left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y}}^{ *} } \right) = \left( {{\text{K}}_{\text{F}}^{ *} + {\text{K}}_{\text{DB}}^{ *} } \right) \cdot {\text{d}}_{\text{y}}^{ *} + \left( {{\text{K}}_{\text{F}}^{ *} + {\text{r}}_{\text{DB}}^{ *} {\text{K}}_{\text{DB}}^{ *} } \right)\left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y}}^{ *} } \right) = {\text{K}}_{\text{F}}^{ *} \left( { 1+\upalpha} \right) \cdot {\text{d}}_{\text{y}}^{ *} + {\text{K}}_{\text{F}}^{ *} \left( { 1+ {\text{r}}_{\text{DB}}^{ *}\upalpha} \right)\left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y}}^{ *} } \right) = {\text{K}}_{\text{F}}^{ *} \cdot \left\{ {{\text{d}}_{\text{pp}}^{ *} +\upalpha \cdot \left[ {{\text{d}}_{\text{y}}^{ *} + {\text{r}}_{\text{DB}}^{ *} \left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y}}^{ *} } \right)} \right]} \right\}$$

that leads to the following result, that is that of Eq. (8):

Fig. 24

Force-displacement behavior of BF system in the case of linear behavior of F

$$\upalpha = \left( {\frac{{{\text{V}}_{\text{PP,BF}}^{ *} }}{{{\text{K}}_{\text{F}}^{ *} }} - {\text{d}}_{\text{pp}}^{ *} } \right) \cdot \frac{ 1}{{{\text{d}}_{\text{y}}^{ *} + {\text{r}}_{\text{DB}}^{ *} \cdot \left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y}}^{ *} } \right)}}\quad q.e.d.$$

Appendix B: Proof of Eq. (9)

Making reference to Fig. 25, the following equalities apply:

$${\text{V}}_{\text{PP,BF}}^{ *} = {\text{K}}_{\text{BF}}^{ *} \cdot {\text{d}}_{\text{y,DB}}^{ *} + {\text{K}}_{\text{BF,py,1}}^{ *} \left( {{\text{d}}_{\text{y,F}}^{ *} - {\text{d}}_{\text{y,DB}}^{ *} } \right) \quad + {\text{K}}_{\text{BF,py,2}}^{ *} \left( {{\text{d}}_{\text{PP}}^{ *} - {\text{d}}_{\text{y,F}}^{ *} } \right) = \left( {{\text{K}}_{\text{F}}^{ *} + {\text{K}}_{\text{DB}}^{ *} } \right) \cdot {\text{d}}_{\text{y,DB}}^{ *}\quad + \left( {{\text{K}}_{\text{F}}^{ *} + {\text{r}}_{\text{DB}}^{ *} {\text{K}}_{\text{DB}}^{ *} } \right)\left( {{\text{d}}_{\text{y,F}}^{ *} - {\text{d}}_{\text{y,DB}}^{ *} } \right) + \left( {{\text{r}}_{\text{F}}^{ *} {\text{K}}_{\text{F}}^{ *} + {\text{r}}_{\text{DB}}^{ *} {\text{K}}_{\text{DB}}^{ *} } \right)\left( {{\text{d}}_{\text{PP}}^{ *} - {\text{d}}_{\text{y,F}}^{ *} } \right) = {\text{K}}_{\text{F}}^{ *} \left( { 1+\upalpha} \right) \cdot {\text{d}}_{\text{y,DB}}^{ *} + {\text{K}}_{\text{F}}^{ *} \left( { 1+ {\text{r}}_{\text{DB}}^{ *}\upalpha} \right)\left( {{\text{d}}_{\text{y,F}}^{ *} - {\text{d}}_{\text{y,DB}}^{ *} } \right) + {\text{K}}_{\text{F}}^{ *} \left( {{\text{r}}_{\text{F}}^{ *} + {\text{r}}_{\text{DB}}^{ *}\upalpha} \right)\left( {{\text{d}}_{\text{PP}}^{ *} - {\text{d}}_{\text{y,F}}^{ *} } \right)$$

that leads to the following result, that is that of Eq. (9):.

Fig. 25

Force-displacement behavior of BF system in the case of non-linear beavior of F

$$\upalpha = \left\{ {\frac{{{\text{V}}_{\text{PP,BF}}^{ *} }}{{{\text{K}}_{\text{F}}^{ *} }} - \left[ {{\text{d}}_{\text{y,F}}^{ *} + {\text{r}}_{\text{F}}^{ *} \left( {{\text{d}}_{\text{PP}}^{ *} - {\text{d}}_{\text{y,F}}^{ *} } \right)} \right]} \right\} \cdot \frac{ 1}{{{\text{d}}_{\text{y,DB}}^{ *} + {\text{r}}_{\text{DB}}^{ *} \cdot \left( {{\text{d}}_{\text{pp}}^{ *} - {\text{d}}_{\text{y,DB}}^{ *} } \right)}}\quad q.e.d.$$

Appendix C: Proof of Eq. (18)

The rheological model of the equivalent DB system can be modeled as springs in series. In Fig. 26 “n” is the total number of storeys, “i” the generic level. The base shear of the equivalent DB system, ${\text{V}}_{{{\text{PP}},{\text{DB}}}}^{*}$, corresponds to the first story shear of the dissipative brace system. Taking this into consideration and making reference to Fig. 27 the following equalities apply:

Fig. 26

Rheological model of DB equivalent system

Fig. 27

Force-displacement behavior of DB system

$${\text{r}}_{\text{DB}}^{ *} = \frac{{{\text{K}}_{\text{DB,py}}^{ *} }}{{{\text{K}}_{\text{DB}}^{ *} }} = \frac{{{\text{V}}_{\text{PP,DB}}^{ *} - {\text{V}}_{\text{y,DB}}^{ *} }}{{{\text{d}}_{\text{PP}}^{ *} - {\text{d}}_{\text{y}}^{ *} }} \cdot \frac{ 1}{{{\text{K}}_{\text{DB}}^{ *} }} = \frac{{{\text{V}}_{\text{PP,DB}} - {\text{V}}_{\text{y,DB}} }}{{{\text{d}}_{\text{PP}} - {\text{d}}_{\text{y}} }} \cdot \frac{ 1}{{{\text{K}}_{\text{DB}}^{ *} }} = \frac{{{\text{V}}_{{{\text{PP,DB,i}} = 1}} - {\text{V}}_{{{\text{y,DB,i}} = 1}} }}{{{\text{d}}_{\text{PP}} - {\text{d}}_{\text{y}} }} \cdot \frac{ 1}{{{\text{K}}_{\text{DB}}^{ *} }} = \frac{{{\text{V}}_{{{\text{y,DB,i}} = 1}} + \left( {\updelta_{{{\text{PP,i}} = 1}} -\updelta_{{{\text{y,i}} = 1}} } \right) \cdot {\text{K}}_{{{\text{DB,py,i}} = 1}} - {\text{V}}_{{{\text{y,DB,i}} = 1}} }}{{{\text{d}}_{\text{PP}} - {\text{d}}_{\text{y}} }} \cdot \frac{ 1}{{{\text{K}}_{\text{DB}}^{ *} }} = \frac{{\left( {\updelta_{{{\text{PP,i}} = 1}} -\updelta_{{{\text{y,i}} = 1}} } \right) \cdot {\text{K}}_{{{\text{DB,py,i}} = 1}} }}{{{\text{d}}_{\text{PP}} - {\text{d}}_{\text{y}} }} \cdot \frac{ 1}{{{\text{K}}_{\text{DB}}^{ *} }}\quad q.e.d.$$